The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

Abstract

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.

A note on convergence results for varying interval valued multisubmeasures

“…In the theory of integration several generalizations have been formulated extending the Riemann and Lebesgue integrals (see for example [6-10, 26, 27, 33] and the references therein). In the present paper we continue the research started in [7,13,20,34] and originating from [24,27]. We provide sufficient conditions for obtaining a Monotone Convergence Theorem for a sequence of interval valued multifunction (G n ) n with respect to interval valued multisubmeasures M and we consider also the asympotic properties of G n dM n for a pair of sequences (G n , M n ) n of interval valued multifunctions and multisubmeasures providing a generalized version of the Monotone Convergence Theorem [22,28,31,41].…”

“…RL interval valued integrability and its properties. In this section, we recall the Riemann-Lebesgue integrability of interval-valued multifunctions with respect to interval-valued multisubmeasures, studied in [13], and whose results are derived from [7], and the properties of this integral that will be usefull for our purpose.…”

“…We want to highlight that the integral, given in Definition 3.2, was obtained with respect to an arbitrary, non negative, set function. For the comparison among this definition and other integrability definitions we can observe that, by [7,Theorem 9] we have that if f is RL integrable with respect to a µ then f is also Birkhoff simple integrable and the two integrals coincide or we refere to the results contained in [7,10,13,24,27].…”

“…For every countable tagged partition Π := {(B n , s n ), n ∈ N} of S, as shown in [13], we have that ) such that for every ε > 0, there exists a countable partition P ε of S, so that for every tagged partition P = {(A n , t n )} n∈N of S with P ≥ P ε , the series σ G,M (P ) is convergent and…”

“…Since G n , G are bounded and µ 2 is of finite variation then, G n , G ∈ RL 1 (M k ) ∩ RL 1 (M ) for every n, k ∈ N; it is a consequence of [7,Proposition 1] and [13,Proposition 2].…”